This is an interesting question, and one which I have given some thought to, as someone who is likely to be teaching world history in the near future. World history, in my interpretation of the field, has three main purposes. Its first, and perhaps original, purpose is to break out of the traditional division of historical study into separate and largely distinct regions. Instead of offering a separate "history of Europe," "history of East Asia," "history of the Middle East," and so forth, world history presents a "connected history" demonstrating the benefits of viewing different regions of the globe in relation to one another. This is done by exploring the economic, cultural, and human linkages tying different regions together. Its second purpose is to demonstrate the degree to which humanity as a whole has a shared history, and common place in the world. Connected histories help illuminate how different regions of the world shaped one another's histories, but world history additionally allows one to zoom out even further and view mankind itself as a unit of analysis. Compartmentalized regional histories do a poor job of presenting and analyzing mankind's place in and relation with the world around us. World history overcomes this by emphasizing globe-spanning phenomena such as climate, the environment, and man's changing relationship with the natural world. This approach taken to the extreme resulted in Big History, a field that takes the universe as its unit of analysis, beginning with the Big Bang and situating mankind into the most zoomed-out perspective possible. The third purpose is to offer a non-Eurocentric alternative to the traditional grand narrative of "Western Civilization." Whether by demonstrating the interconnected nature of the world (and thus undermining more extreme claims of European exceptionalism), or by zooming out to view mankind as a whole, Europe stops being the center of attention and becomes one out of many world regions to be studied and understood.

These three goals are all things to keep in mind when designing a world history course. The thing to be avoided is something like what has been called by feminist historians the "add women and stir" approach to history: in our case simply taking the traditional Western Civ narrative, adding a few other regions like China and India, and calling it a history of the world. The reason this often happens is that no coherent master narrative of world history has become widely accepted to replace that of Western Civ, aside from the notion that the world became increasingly interconnected over time (or the above-mentioned Big History, which some have taken up in their teaching), leaving us either with this kind of Western Civ Frankenstein or with fragmented serial regional histories taught by teachers who haven't necessarily had training in how to connect them together into a real world history.

"Here is the crux of world history's distinctive teaching and, consequently, learning problem. When confronted with lists of US or Western history standards or corresponding textbooks stuffed with details, most teachers appear to have a rough sense of key turning points and major events - a broad narrative on which they can hang the details. To use Ladurie's categorization scheme of historians, when approaching more familiar national or regional histories most teachers are like parachutists gliding over the historical landscape. From that distance they can see the entire temporal and spatial topography and the relationships among large structures; and while floating down they are able to situate close-up detail as it comes into view. However, these teachers do not seem to approach world history from the grand perspective that distance allows, acting instead like Ladurie's truffle hunters, noses in the dirt sniffing out disconnected but precious details."

Robert B. Bain, "Challenges of Teaching and Learning World History," in A Companion to World History, 1st ed. Edited by Douglas Northrop (Blackwell Publishing Ltd., 2015), pp. 113-4.

Yet world history is all about the big picture, and being able to view the topic from a distance is very important.

About the new AP regulations: I do not believe that a course's chronological limits determine whether or not it is Eurocentric. At the university level, world history is already typically divided into pre- and post-1500(ish) courses, and the latter hardly has to be only "colonial world history." The great majority of the world, after all, was not under European colonial rule at that time, and would not be for hundreds of years (nor does being under colonial rule erase those peoples' agency). What matters most is how that history is presented, not what time period is chosen. The 1450/1492/1500 dividing line has a long history in the "Rise of the West" master narrative, but that doesn't mean that instructors need to teach this period from that perspective, on the contrary! Rather than centering the story around European explorers, conquistadors, and missionaries "discovering the world," this can be made just one part of the story of the widening network of Afro-Eurasian interconnectivity which predated Europe's voyages of discovery by centuries. So a post-1500 world history still can accomplish our purpose #3, and purpose #1 is best served by a strong post-1500 focus as well - most studies of world history are focused on that period, when global interconnections became increasingly intense, despite having a long history before then as well. As I see it, the greatest danger of adding a chronological limitation to the teaching of world history lies in how it undermines instructors' abilities to present human history as a united whole, our purpose #2 of world history.

However, this is making the best out of a bad situation. I don't think cutting out all world history prior to 1450 is a good thing, but I don't think it has to mean a return to Eurocentrism either. So to finally return to your question:

What is the "history of the world" that everyone should be at least cursory aware of?

There is no one answer, of course. In my view world history should make students aware of how humanity evolved and spread across the earth, how humans divided into different ways of life (pastoral, agricultural, hunter-gatherer, etc.), and how what we call "civilization" came about. They should be taught about how the environment shaped the human experience. Broadly, they should be taught about the features which pre-modern agricultural societies mostly had in common: the nature of inequality, economic activity, the life cycle, the state and political authority. Johannes Preiser-Kapeller wrote a wonderful world history of late antiquity which captures what I consider to be important: The appearance and dissolution of empires (not by exploring their individual histories, but empire as a historical phenomenon), the propagation of cultural traditions among trans-regional elites, the emergence and transmission of religious traditions, the spread of diasporas and human communities across the continents, the movement of trade goods, and the way changes in the Earth's climate impacted human societies. It is less important that students learn the details of what events were occurring in any given place at any given time, than that they come to see the world as a single world, recognizing global processes, continuities and discontinuities, and making comparisons. The details they can get from textbooks, and from what their instructor chooses to focus on, which will be necessarily tied to his or her own strengths. This is what world history really is, after all: not a history of everything, like a watered-down conglomeration of all the different regional histories. Rather, a shift of perspective away from the regional, embracing phenomena with global significance.

Not to discourage anyone, and there's certainly a lot that could be said here, but u/agentdcf had an excellent post covering the traditional Western Civ narrative and it's problems here.

A bit of clarification: are you asking "given complete freedom in course design, what would be the best way to teach world history?" or "what would be the best way to teach world history within the new AP constraints?"